3.18 \(\int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\)

Optimal. Leaf size=291 \[ \frac {2 b C \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} (a C h-b B h+b C g) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {h (d e-c f)}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

2*b*C*EllipticE(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f
*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2)/d/h/f^(1/2)/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)-2*(-B*b*h+C*a*h+C
*b*g)*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f
*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)/d/h/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)

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Rubi [A]  time = 0.28, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {24, 158, 114, 113, 121, 120} \[ \frac {2 b C \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} (a C h-b B h+b C g) F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}} \]

Antiderivative was successfully verified.

[In]

Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*b*C*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x
])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g -
c*h)]) - (2*Sqrt[-(d*e) + c*f]*(b*C*g - b*B*h + a*C*h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*
g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*S
qrt[f]*h*Sqrt[e + f*x]*Sqrt[g + h*x])

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-((b*c - a*d)/d), 0]

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &
& SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-((b*c - a*d)/d)] || NegQ[-((b*e - a*f)/f)
])

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 158

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rubi steps

\begin {align*} \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\frac {\int \frac {b^2 (b B-a C)+b^3 C x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2}\\ &=\frac {(b C) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{h}-\frac {(b C g-b B h+a C h) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{h}\\ &=-\frac {\left ((b C g-b B h+a C h) \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}} \, dx}{h \sqrt {e+f x}}+\frac {\left (b C \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x}\right ) \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}} \, dx}{h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}\\ &=\frac {2 b C \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {\left ((b C g-b B h+a C h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}} \, dx}{h \sqrt {e+f x} \sqrt {g+h x}}\\ &=\frac {2 b C \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {-d e+c f} (b C g-b B h+a C h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}\\ \end {align*}

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Mathematica [C]  time = 2.48, size = 326, normalized size = 1.12 \[ \frac {2 \left (-i d h (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} (a C f-b B f+b C e) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d e}{f}-c}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )+b C d^2 (e+f x) (g+h x) \sqrt {\frac {d e}{f}-c}+i b C h (c+d x)^{3/2} (d e-c f) \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d e}{f}-c}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )\right )}{d^2 f h \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \sqrt {\frac {d e}{f}-c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*(b*C*d^2*Sqrt[-c + (d*e)/f]*(e + f*x)*(g + h*x) + I*b*C*(d*e - c*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f
*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g -
c*f*h)/(d*e*h - c*f*h)] - I*d*(b*C*e - b*B*f + a*C*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqr
t[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h -
c*f*h)]))/(d^2*Sqrt[-c + (d*e)/f]*f*h*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])

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fricas [F]  time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C b x - C a + B b\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{d f h x^{3} + c e g + {\left (d f g + {\left (d e + c f\right )} h\right )} x^{2} + {\left (c e h + {\left (d e + c f\right )} g\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fric
as")

[Out]

integral((C*b*x - C*a + B*b)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)/(d*f*h*x^3 + c*e*g + (d*f*g + (d*e + c*
f)*h)*x^2 + (c*e*h + (d*e + c*f)*g)*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac
")

[Out]

integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

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maple [B]  time = 0.03, size = 673, normalized size = 2.31 \[ \frac {2 \left (B b c d f h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-B b \,d^{2} e h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-C a c d f h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+C a \,d^{2} e h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-C b \,c^{2} f h \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+C b c d e h \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+C b c d f g \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-C b c d f g \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-C b \,d^{2} e g \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+C b \,d^{2} e g \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )\right ) \sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \sqrt {-\frac {\left (h x +g \right ) d}{c h -d g}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {d x +c}}{\left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right ) d^{2} f h} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)

[Out]

2*(B*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c*d*f*h-B*EllipticF(((d*x+c)/(c*
f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*d^2*e*h-C*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/
(c*h-d*g)/f*h)^(1/2))*a*c*d*f*h+C*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*d^2
*e*h-C*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c*d*f*g+C*EllipticF(((d*x+c)/(
c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*d^2*e*g-C*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e
)/(c*h-d*g)/f*h)^(1/2))*b*c^2*f*h+C*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c
*d*e*h+C*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c*d*f*g-C*EllipticE(((d*x+c)
/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*d^2*e*g)*(-(f*x+e)/(c*f-d*e)*d)^(1/2)*(-(h*x+g)/(c*h-d*
g)*d)^(1/2)*((d*x+c)/(c*f-d*e)*f)^(1/2)/h/f/d^2*(h*x+g)^(1/2)*(f*x+e)^(1/2)*(d*x+c)^(1/2)/(d*f*h*x^3+c*f*h*x^2
+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxi
ma")

[Out]

integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {-C\,a^2+B\,a\,b+C\,b^2\,x^2+B\,b^2\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*b^2*x^2 - C*a^2 + B*a*b + B*b^2*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)*(c + d*x)^(1/2)),x)

[Out]

int((C*b^2*x^2 - C*a^2 + B*a*b + B*b^2*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)*(c + d*x)^(1/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Timed out

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